Isothermal Turbulence Research Articles

THE DENSITY VARIANCE–MACH NUMBER RELATION IN SUPERSONIC, ISOTHERMAL TURBULENCE

We examine the relation between the density variance and the mean-square Mach number in supersonic, isothermal turbulence, assumed in several recent analytic models of the star formation process. From a series of calculations of supersonic, hydrodynamic turbulence driven using purely solenoidal Fourier modes, we find that the `standard' relationship between the variance in the log of density and the Mach number squared, i.e., sigma^2_(ln rho/rhobar)=ln (1+b^2 M^2), with b = 1/3 is a good fit to the numerical results in the supersonic regime up to at least Mach 20, similar to previous determinations at lower Mach numbers. While direct measurements of the variance in linear density are found to be severely underestimated by finite resolution effects, it is possible to infer the linear density variance via the assumption of log-normality in the Probability Distribution Function. The inferred relationship with Mach number, consistent with sigma_(rho/rhobar) ~ b M with b=1/3, is, however, significantly shallower than observational determinations of the relationship in the Taurus Molecular Cloud and IC5146 (both consistent with b~ 0.5), implying that additional physics such as gravity is important in these clouds and/or that turbulent driving in the ISM contains a significant compressive component. Magnetic fields are not found to change this picture significantly, in general reducing the measured variances and thus worsening the discrepancy with observations.

ON THE DENSITY DISTRIBUTION IN STAR-FORMING INTERSTELLAR CLOUDS

We use deep adaptive mesh refinement simulations of isothermal self-gravitating supersonic turbulence to study the imprints of gravity on the mass density distribution in molecular clouds. The simulations show that the density distribution in self-gravitating clouds develops an extended power-law tail at high densities on top of the usual lognormal. We associate the origin of the tail with self-similar collapse solutions and predict the power index values in the range from -7/4 to -3/2 that agree with both simulations and observations of star-forming molecular clouds.

Highly-resolved LES and PIV Analysis of Isothermal Turbulent Opposed Jets for Combustion Applications

Turbulent opposed jet (TOJ) burners are an interesting test case for fundamental combustion research and a good benchmark for the available modelling approaches. However, these opposed jet flames strongly depend on the turbulence generation inside the nozzle, which is usually achieved through a perforated plate upstream of the nozzle exit. The present work investigates the flow from these perforated plates and the subsequent turbulence generation in great detail. We present results from highly-resolved large eddy simulations (LES) of the in-nozzle flow in turbulent opposed jets alongside state-of-the-art particle image velocimetry (PIV) at standard and high repetition rates taken inside a glass nozzle. The in-nozzle PIV data provides the LES inflow conditions with unprecedented detail, which are used to follow the initial jet development behaviour known from PIV, before jet coalescence, turbulence production and decay further downstream in the nozzles are successfully predicted. In regions where the PIV experiment suffers from inherent limitations like reflections and the velocity bias, the LES data is available to still obtain a detailed picture of the flow. The sensitivity of the simulations to various physical and numerical parameters is discussed in detail. Results from LES and PIV are compared qualitatively and quantitatively in terms of first and second moments of velocity, temporal autocorrelations, and energy density spectra. Significant deviations are found in the frequency (20%) and strength of vortex shedding from the inlet plane only, whereas the qualitative and quantitative agreement between simulation and experiment is otherwise excellent throughout, implying that a successful large eddy simulation of a turbulent opposed jet can be performed in a domain that includes the perforated plates.

MIXING IN SUPERSONIC TURBULENCE

In many astrophysical environments, mixing of heavy elements occurs in the presence of a supersonic turbulent velocity field. Here we carry out the first systematic numerical study of such passive scalar mixing in isothermal supersonic turbulence. Our simulations show that the ratio of the scalar mixing timescale, $\tau_{\rm c}$, to the flow dynamical time, $\tau_{\rm dyn}$ (defined as the flow driving scale divided by the rms velocity), increases with the Mach number, $M$, for $M \lsim3$, and becomes essentially constant for $M \gsim3.$ This trend suggests that compressible modes are less efficient in enhancing mixing than solenoidal modes. However, since the majority of kinetic energy is contained in solenoidal modes at all Mach numbers, the overall change in $\tau_{\rm c}/\tau_{\rm dyn}$ is less than 20\% over the range $1 \lsim M \lsim 6$. At all Mach numbers, if pollutants are injected at around the flow driving scale, $\tau_{\rm c}$ is close to $\tau_{\rm dyn}.$ This suggests that scalar mixing is driven by a cascade process similar to that of the velocity field. The dependence of $\tau_{\rm c}$ on the length scale at which pollutants are injected into flow is also consistent with this cascade picture. Similar behavior is found for the variance decay timescales for scalars without continuing sources. Extension of the scalar cascade picture to the supersonic regime predicts a relation between the scaling exponents of the velocity and the scalar structure functions, with the scalar structure function becoming flatter as the velocity scaling steepens with Mach number. Our measurements of the volume-weighted velocity and scalar structure functions confirm this relation for $M\lsim 2,$ but show discrepancies at $M \gsim 3$.

A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence

We compare the statistics of driven, supersonic turbulence at high Mach number using FLASH a widely used Eulerian grid-based code and PHANTOM, a Lagrangian SPH code at resolutions of up to 512^3 in both grid cells and SPH particles. We find excellent agreement between codes on the basic statistical properties: a slope of k^-1.95 in the velocity power spectrum for hydrodynamic, Mach 10 turbulence, evidence in both codes for a Kolmogorov-like slope of k^-5/3 in the variable rho^1/3 v as suggested by Kritsuk et al. and a log-normal PDF with a width that scales with Mach number and proportionality constant b=0.33-0.5 in the density variance-Mach number relation. The measured structure function slopes are not converged in either code at 512^3 elements. We find that, for measuring volumetric statistics such as the power spectrum slope and structure function scaling, SPH and grid codes give roughly comparable results when the number of SPH particles is approximately equal to the number of grid cells. In particular, to accurately measure the power spectrum slope in the inertial range, in the absence of subgrid models, requires at least 512^3 computational elements in either code. On the other hand the SPH code was found to be better at resolving dense structures, giving max. densities at a resolution of 128^3 particles that were similar those resolved in the grid code at 512^3 cells, reflected also in the high density tail of the PDF. We find SPH to be more dissipative at comparable numbers of computational elements in statistics of the velocity field, but correspondingly less dissipative than the grid code in the statistics of density weighted quantities such as rho^1/3 v. For SPH simulations of high Mach number turbulence we find it important to use sufficient non-linear beta-viscosity to prevent particle interpenetration in shocks (we require beta=4 instead of the default beta=2).

Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence

We investigate the influence of the turbulence forcing on the mass distributions of gravitationally unstable cores by postprocessing data from simulations of non-selfgravitating isothermal supersonic turbulence with varying resolution. In one set of simulations solenoidal forcing is applied, while the second set uses purely compressive forcing to excite turbulent motions. From the resulting density field, we compute the mass distribution of gravitationally unstable cores by means of a clump-finding algorithm. Using the time-averaged probability density functions of the mass density, semi-analytic mass distributions are calculated from analytical theories. We apply stability criteria that are based on the Bonnor-Ebert mass resulting from the thermal pressure and from the sum of thermal and turbulent pressure. Although there are uncertainties in the application of the clump-finding algorithm, we find systematic differences in the mass distributions obtained from solenoidal and compressive forcing. Compressive forcing produces a shallower slope in the high-mass power-law regime compared to solenoidal forcing. The mass distributions also depend on the Jeans length resulting from the choice of the mass in the computational box, which is freely scalable for non-selfgravitating isothermal turbulence. Provided that all cores are numerically resolved and most cores are small compared to the length scale of the forcing, the normalised core mass distributions are found to be close to the semi-analytic models. Especially for the high-mass tails, the Hennebelle-Chabrier theory implies that the additional support due to turbulent pressure is important.

Magnetic Fields in Molecular Clouds

AbstractSupersonic magneto-hydrodynamic (MHD) turbulence in molecular clouds (MCs) plays an important role in the process of star formation. The effect of the turbulence on the cloud fragmentation process depends on the magnetic field strength. In this work we discuss the idea that the turbulence is super-Alfvénic, at least with respect to the cloud mean magnetic field. We argue that MCs are likely to be born super-Alfvénic. We then support this scenario based on a recent simulation of the large-scale warm interstellar medium turbulence. Using small-scale isothermal MHD turbulence simulation, we also show that MCs may remain super-Alfvénic even with respect to their rms magnetic field strength, amplified by the turbulence. Finally, we briefly discuss the comparison with the observations, suggesting that super-Alfvénic turbulence successfully reproduces the Zeeman measurements of the magnetic field strength in dense MC clouds.

THE FRACTAL DENSITY STRUCTURE IN SUPERSONIC ISOTHERMAL TURBULENCE: SOLENOIDAL VERSUS COMPRESSIVE ENERGY INJECTION

In a systematic study, we compare the density statistics in high-resolution numerical experiments of supersonic isothermal turbulence, driven by the usually adopted solenoidal (divergence-free) forcing and by compressive (curl-free) forcing. We find that for the same rms Mach number, compressive forcing produces much stronger density enhancements and larger voids compared to solenoidal forcing. Consequently, the Fourier spectra of density fluctuations are significantly steeper. This result is confirmed using the {delta}-variance analysis, which yields power-law exponents {beta} {approx} 3.4 for compressive forcing and {beta} {approx} 2.8 for solenoidal forcing. We obtain fractal dimension estimates from the density spectra and {delta}-variance scaling, and by using the box counting, mass size, and perimeter area methods applied to the volumetric data, projections, and slices of our turbulent density fields. Our results suggest that compressive forcing yields fractal dimensions significantly smaller compared to solenoidal forcing. However, the actual values depend sensitively on the adopted method, with the most reliable estimates based on the {delta}-variance, or equivalently, on Fourier spectra. Using these methods, we obtain D {approx} 2.3 for compressive and D {approx} 2.6 for solenoidal forcing, which is within the range of fractal dimension estimates inferred from observations (D {approx} 2.0-2.7). The velocity dispersion tomore » size relations for both solenoidal and compressive forcings obtained from velocity spectra follow a power law with exponents in the range 0.4-0.5, in good agreement with previous studies.« less

Numerical simulations of compressively driven interstellar turbulence

Context. Supersonic turbulence is ubiquitous in the interstellar me dium and plays an important role in contemporary star formation. Aims. To perform a high-resolution numerical simulation of supersonic isothermal turbulence driven by compressive large-scale forcing and to analyse various statistical properties. Methods. The compressible Euler equations with an external stochastic force field dominated by rotation-free modes are solved wi th the piecewise parabolic method. Both a static grid and adaptive mesh refinement is used with an effective resolution N = 768 3 . Results. After a transient phase dominated by shocks, turbulence evolves into a steady state with root mean square Mach number ≈ 2.2... 2.5, in which cloud-like structures of over-dense gas are surr ounded by highly rarefied gas. The index of the turbulence ene rgy spectrum functionβ≈ 2.0 in the shock-dominated phase. As the flow approaches statis tical equilibrium, the spectrum flattens, with β≈ 1.9. For the scaling exponent of the root mean square velocity fl uctuation, we obtainγ≈ 0.43 from the velocity structure functions of second order. These results are well within the range of observed scaling properties for the velocity dispersion in molecular clouds. Calculating structure functions of order p = 1,..., 5, we find for all scaling exponents significant deviations from the Kolmogor ovBurgers model proposed by Boldyrev. Our results are very well described by a general log-Poisson model with a higher degree of intermittency, which implies an influence of the forcing on t he scaling properties. The spectral index of the quadratic l ogarithmic density fluctuation βδ≈ 1.8. Contrary to previous numerical results for isothermal turbulence, we obtain a skewed probability density function of the mass density fluctuations that is not consist ent with log-normal statistics and entails a substantially higher fraction of mass in the density peaks than implied by the Padoan-Nordlund relation between the variance of the density fluctuation s and the Mach number. Conclusions. Even putting aside further complexity due to magnetic fields , gravity or thermal processes, we question the notion that Larson-type relations are a consequence of universal supersonic turbulence scaling. For a genuine understanding, it seems necessary to account for the production mechanism of turbulence in the ISM.

The Turbulent Warm Ionized Medium: Emission Measure Distribution and MHD Simulations

We present an analysis of the distribution of H-alpha emission measures for the warm ionized medium (WIM) of the Galaxy using data from the Wisconsin H-Alpha Mapper (WHAM) Northern Sky Survey. Our sample is restricted to Galactic latitudes |b| > 10. We removed sightlines intersecting nineteen high-latititude classical H II regions, leaving only sightlines that sample the diffuse WIM. The distribution of EM sin |b| for the full sample is poorly characterized by a single normal distribution, but is extraordinarily well fit by a lognormal distribution, with = 0.146 +/- 0.001 and standard deviation 0.190 +/- 0.001. drops from 0.260 +/- 0.002 at Galactic latitude 10<|b|<30 to 0.038 +/- 0.002 at Galactic latitude 60<|b|<90. The distribution may widen slightly at low Galactic latitude. We compare the observed EM distribution function to the predictions of three-dimensional magnetohydrodynamic simulations of isothermal turbulence within a non-stratified interstellar medium. We find that the distribution of EM sin |b| is well described by models of mildy supersonic turbulence with a sonic Mach number of ~1.4-2.4. The distribution is weakly sensitive to the magnetic field strength. The model also successfully predicts the distribution of dispersion measures of pulsars and H-alpha line profiles. In the best fitting model, the turbulent WIM occupies a vertical path length of 400-500 pc within the 1.0-1.8 kpc scale height of the layer. The WIM gas has a lognormal distribution of densities with a most probable electron density n_{pk} = 0.03 cm^{-3}. We also discuss the implications of these results for interpreting the filling factor, the power requirement, and the magnetic field of the WIM.

Analytical Theory for the Initial Mass Function: CO Clumps and Prestellar Cores

We derive an analytical theory of the prestellar core initial mass function (IMF) based on an extension of the Press-Schechter statistical formalism. Our approach relies on the general concept of the gravothermal and gravoturbulent collapse of a molecular cloud, with a selection criterion based on the thermal or turbulent Jeans mass, which yields the derivation of the mass spectrum of self-gravitating objects in a quiescent or a turbulent environment. The same formalism also yields the mass spectrum of non-self-gravitating clumps produced in supersonic flows. The mass spectrum of the self-gravitating cores reproduces well the observed IMF. The theory predicts that the shape of the IMF results from two competing contributions, namely, a power law at large scales and an exponential cutoff (lognormal form) centered around the characteristic mass for gravitational collapse. The cutoff exists both in the case of thermal or turbulent collapse, provided that the underlying density field has a lognormal distribution. Whereas pure thermal collapse produces a power-law tail steeper than the Salpeter value, -->dN/dlog M M−x with -->x 1.35, the latter is recovered exactly for the (three-dimensional) value of the spectral index of the velocity power spectrum, -->n 3.8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that -->x = (n + 1)/(2n − 4) for self-gravitating structures and -->x = 2 − n'/3 for non-self-gravitating structures, where -->n' is the power spectrum index of -->log ρ . We show that, whereas supersonic turbulence promotes the formation of both massive stars and brown dwarfs, it has an overall negative impact on star formation, decreasing the star formation efficiency. This theory provides a novel theoretical foundation to understand the origin of the IMF and provides useful guidance to numerical simulations exploring star formation, while making testable predictions.

Effects of Free-stream Turbulence on a Turbulent Boundary Layer (1st Report, Effects of Grid Turbulence on an Isothermal Turbulent Boundary Layer)

Effects of external turbulence in a free stream on an isothermal turbulent boundary layer over a flat boundary are investigated by means of laboratory experiments using a wind tunnel. A turbulence-generating grid is installed upstream of the turbulent boundary layer to generate nearly isotropic, external turbulence having 1.3-2.4% turbulence levels. Instantaneous velocities, U and V, are measured using a hot wire anemometry with an X-probe. The results show that the log-law for the time-averaged mean velocity holds for turbulent boundary layer with external grid turbulence. However, turbulence intensities and the Reynolds stress are strongly suppressed by the external grid turbulence under the present conditions. The suppressions are caused by the decrease of production terms in the respective transport equations. The power spectra of u, v and the cospectra of u and v show that almost all scale motions are suppressed by the external grid turbulence.

Turbulent thermophoresis effect on particle transport processes

The analytical solutions have been obtained with the aid of the surface rejuvenation model for the quantitative predictions of mean eddy approach distance, mean thermophoretic velocity and mean thermophoretic transport rate in detail. The mean sublayer residence time solved in accordance with the velocity distribution in the fully turbulent fluid flow is consistent with visual observations. The local cumulative submicron particle transport rate has been theoretically predicted including the conjoint effects of eddy diffusion, Brownian diffusion, and thermophoretic effect. The results give an explanation for the complex phenomena of the comparable coupling between the thermal and turbulent mechanism during the time intervals of successive eddies, especially for the heating surface. A good agreement of dimensionless mean transport rate in an isothermal turbulence fluid flow proves the subsequent developments of the coupling on the particle transfer mechanisms in the viscous sublayer.

A General Theory of Turbulence‐regulated Star Formation, from Spirals to Ultraluminous Infrared Galaxies

We derive an analytic prediction for the star formation rate in environments ranging from normal galactic disks to starbursts and ULIRGs in terms of the observables of those systems. Our calculation is based on three premises: (1) star formation occurs in virialized molecular clouds that are supersonically turbulent; (2) the density distribution within these clouds is lognormal, as expected for supersonic isothermal turbulence; and (3) stars form in any subregion of a cloud that is so overdense that its gravitational potential energy exceeds the energy in turbulent motions. We show that a theory based on this model is consistent with simulations and with the observed star formation rate in the Milky Way. We use our theory to derive the Kennicutt-Schmidt law from first principles and make other predictions that can be tested by future observations. We also provide an algorithm for estimating the star formation rate that is suitable for inclusion in numerical simulations.

Density Power Spectrum of Compressible Hydrodynamic Turbulent Flows

Turbulent flows are ubiquitous in astrophysical environments, and understanding density structures and their statistics in turbulent media is of great importance in astrophysics. In this paper, we study the density power spectra, $P_{\rho}$, of transonic and supersonic turbulent flows through one and three-dimensional simulations of driven, isothermal hydrodynamic turbulence with root-mean-square Mach number in the range of $1 \la M_{\rm rms} \la 10$. From one-dimensional experiments we find that the slope of the density power spectra becomes gradually shallower as the rms Mach number increases. It is because the density distribution transforms from the profile with {\it discontinuities} having $P_{\rho} \propto k^{-2}$ for $M_{\rm rms} \sim 1$ to the profile with {\it peaks} having $P_{\rho} \propto k^0$ for $M_{\rm rms} \gg 1$. We also find that the same trend is carried to three-dimension; that is, the density power spectrum flattens as the Mach number increases. But the density power spectrum of the flow with $M_{\rm rms} \sim 1$ has the Kolmogorov slope. The flattening is the consequence of the dominant density structures of {\it filaments} and {\it sheets}. Observations have claimed different slopes of density power spectra for electron density and cold H I gas in the interstellar medium. We argue that while the Kolmogorov spectrum for electron density reflects the {\it transonic} turbulence of $M_{\rm rms} \sim 1$ in the warm ionized medium, the shallower spectrum of cold H I gas reflects the {\it supersonic} turbulence of $M_{\rm rms} \sim$ a few in the cold neutral medium.

Attenuation Caused by a Distant Isothermal Turbulent Screen

We analyze in detail the attenuation caused by an isothermal turbulent distant foreground dust screen. The attenuation curve is well determined by two parameters, the absolute-to-relative attenuation ratio R = AV/E(B - V) and the absolute attenuation AV. We show quantitatively how these two observable quantities depend on the statistical properties of the local density and the mean attenuation AV and how they vary with the thickness of the screen measured in units of the largest turbulent scale. The attenuation through a turbulent medium is characterized by higher transparency and a flatter attenuation curve in comparison with a homogeneous dust screen. In general, the effect of a turbulent medium on the attenuation increases with slice thickness. In the limit of thick slices, typically larger than the maximum turbulent scale, R asymptotically approaches a maximum value.

Scaling Relations for Turbulence in the Multiphase Interstellar Medium

We employ a generalization of the She & Lévêque model to study velocity scaling relations based on our simulations of thermal instability-induced turbulence. Being a by-product of the interstellar phase transition, such multiphase turbulence tends to be more intermittent than compressible isothermal turbulence. Due to radiative cooling, which promotes nonlinear instabilities in supersonic flows, the Hausdorff dimension of the most singular dissipative structures, D, can be as high as 2.3, while in supersonic isothermal turbulence D is limited by shock dissipation to D ⩽ 2. We also show that single-phase velocity statistics carry only incomplete information on the turbulent cascade in a multiphase medium. We briefly discuss the possible implications of these results on the hierarchical structure of molecular clouds and on star formation.

Hydrodynamical Simulations of Molecular Dynamics in Supersonic Turbulent Flow

Here we present results from simulations of turbulence in star forming environments obtained by coupling three-dimensional hydrodynamical models with appropriate chemical processes. We investigate regimes of decaying high-speed molecular turbulence. Here we analyse PDFs of density for the volume, mass, molecular mass and the energy distribution over the range of scales. We compare our results to those previously obtained for isothermal turbulence and suggest possible explanations.

The correlation between magnetic pressure and density in compressible MHD turbulence

We study, both analytically and numerically, the behavior of magnetic pressure and density fluctuations in turbulent isothermal magnetohydrodynamic (MHD) flows in a slab geometry. We first consider “simple” MHD waves, which are the nonlinear analogue of the slow, fast and Alfvén linear waves, and show that the dependence of magnetic field strength B on density ρ in a simple wave depends on the mode which is considered: for the slow mode, , while for the fast mode, . We also perform a perturbative analysis about a circularly-polarized plane Alfvén wave to investigate Alfvén wave pressure, recovering the results of McKee and Zweibel that , with at large Ma, at moderate Ma and long wavelengths, and at low Ma. This wide variety of behaviors implies that a single polytropic description of magnetic pressure is not possible in general, but instead depends on which mode dominates the density fluctuation production. This in turn depends on the angle θ between the magnetic field and the direction of wave propagation and on the Alfvénic Mach number Ma. Typically, at small Ma, the slow mode dominates, and B is anticorrelated with ρ . At large Ma, both modes contribute to density fluctuation production, and the magnetic pressure decorrelates from density, exhibiting a large scatter, which however decreases towards higher densities. In this case, magnetic “pressure” does not act as a restoring force, but rather as a random forcing. These results have implications for the probability density function (PDF) of mass density. The non-systematic behavior of the magnetic pressure causes the PDF to maintain the log-normal shape corresponding to non-magnetic isothermal turbulence, except in cases where the slow mode dominates, in which the PDF develops an excess at low densities because the magnetic “random forcing” becomes density-dependent. Our results are consistent with the low values and apparent lack of correlation between the magnetic field strength and density in surveys of the lower-density molecular gas, and also with the recorrelation apparently seen at higher densities, if the Alfvénic Mach number is relatively large there.

Hydrodynamical simulations of the decay of high-speed molecular turbulence — I. Dense molecular regions

We present the results from three-dimensional hydrodynamical simulations of decaying high-speed turbulence in dense molecular clouds. We compare our results, which include a detailed cooling function, molecular hydrogen chemistry and a limited C and O chemistry, with those previously obtained for decaying isothermal turbulence. After an initial phase of shock formation, power-law decay regimes are uncovered, as in the isothermal case. We find that the turbulence decays faster than in the isothermal case because the average Mach number remains higher, owing to the radiative cooling. The total thermal energy, initially raised by the introduction of turbulence, decays only a little more slowly than the kinetic energy. We discover that molecule reformation, as the fast turbulence decays, is several times faster than that predicted for a non-turbulent medium. This is caused by moderate speed shocks which sweep through a large fraction of the volume, compressing the gas and dust. Through reformation, the molecular density and molecular column appear as complex patterns of filaments, clumps and some diffuse structure. In contrast, the molecular fraction has a wider distribution of highly distorted clumps and copious diffuse structure, so that density and molecular density are almost identically distributed during the reformation phase. We conclude that molecules form in swept-up clumps but effectively mix throughout via subsequent expansions and compressions.